Giving Mesh thickness with additional requirements
I don't think you can use a modifier in this case. You will probably have to do it manually. 


Go into edit mode tab
select all a
extrude e
scale shift+y, (which will only scale on x and z.)


Now, if there are faces which are on top of each other (in the same place), you can try any of these options;


s, then y to scale along y.
Change pivot point to individual origins, then scale on y.
Simply adjust it manually.


It's hard to say what you should do, because I haven't seen the mesh, but this should help. Keep in mind, that I am assuming facing down your tunnel is the y axis. When scaling, you want to exclude the axis you are looking down, so that it only gets wider, not longer.

I hope this helped, please comment with any questions.